On polynomial invariants of several qubits
Abstract
It is a recent observation that entanglement classification for qubits is closely related to local SL(2,)-invariants including the invariance under qubit permutations, which has been termed SL* invariance. In order to single out the SL* invariants, we analyze the SL(2,)-invariants of four resp. five qubits and decompose them into irreducible modules for the symmetric group S4 resp. S5 of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of SL*-invariants vanishing on arbitrary product states. We find that low degree homogeneous components of this ideal can be constructed in full by using the approach introduced in [Phys. Rev. A 72, 012337 (2005)]. Our analysis highlights an intimate connection between this latter procedure and the standard methods to create invariants, such as the -process. As the degrees of invariants increase, the alternative method proves to be particularly efficient.